A theorem of Pólya on polynomials

Abstract

Among the many contributions of Pólya to analysis, the following has always been Erdós’ favorite, both for the surprising result and for the beauty of its proof. Suppose thatEquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI % cacaWG6bGaaiykaiabg2da9iaadQhadaahaaWcbeqaaiaad6gaaaGc % cqGHRaWkcaWGIbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaki % abgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadkgadaWgaaWcbaGa % aGimaaqabaaaaa!46A7!]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f(z) = {z^n} + {b_{n - 1}} + ... + {b_0}$$ is a complex polynomial of degree n ≥ 1 with leading coefficient 1. Associate with f (z)the setEquationSource MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2 % da9iaacUhacaWG6bGaeyicI4Saam4qaiaacQdadaabdaqaaiaadAga % caGGOaGaamOEaiaacMcacqGHKjYOcaaIYaaacaGLhWUaayjcSdGaai % yFaiaacYcaaaa!4751!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$% C = \{ z \in C:\left| {f(z) \leqslant 2} \right|\} ,$$ that is, C is the set of points which are mapped under f into the circle of radius 2 around the origin in the complex plane. So for n = 1 the domain C is just a circular disk of diameter 4.